A rubber ball dropped from a height of exactly 5 ft bounces (hits the floor) several times, losing 10% of its kinetic energy each bounce. After how many bounces will the ball subsequently not rise above 2 ft?
To solve this problem, we need to understand the concept of kinetic energy and how it relates to the height of the ball.
The kinetic energy of an object in motion is given by the formula:
KE = 1/2 * m * v^2
where KE is the kinetic energy, m is the mass of the object, and v is its velocity.
Assuming the mass of the ball remains constant throughout the bounces, we can ignore it for simplicity. Therefore, we can say that the kinetic energy is directly related to the square of the velocity of the ball.
Now, let's break down the problem step by step:
1. You mentioned that the ball loses 10% of its kinetic energy with each bounce. This means the velocity of the ball reduces by a factor of 10% with each bounce.
2. Initially, the ball is dropped from a height of 5 ft, which means it will have a certain velocity at the moment of impact with the floor.
3. After hitting the floor, the ball bounces back upwards. Due to the loss in kinetic energy, it will not reach the same height as its initial starting position.
4. To determine when the ball will subsequently not rise above 2 ft, we need to find out how many bounces it takes for the ball's maximum height to drop below 2 ft.
Now, let's calculate the maximum height of the ball after each bounce:
1st bounce: The ball reaches a maximum height when all the initial kinetic energy has been converted into potential energy. Since the ball loses 10% of its kinetic energy, its maximum height will also be reduced by 10%. Therefore, the ball reaches a maximum height of 5 ft * 90% = 4.5 ft.
2nd bounce: The ball loses 10% of its kinetic energy again, resulting in a maximum height of 4.5 ft * 90% = 4.05 ft.
3rd bounce: Following the same pattern, the maximum height is 4.05 ft * 90% = 3.645 ft.
4th bounce: Maximum height = 3.645 ft * 90% = 3.2805 ft.
5th bounce: Maximum height = 3.2805 ft * 90% = 2.95245 ft.
At this point, the maximum height of the ball is below 2 ft (which is 1.90 ft). Therefore, after the 5th bounce, the ball subsequently will not rise above 2 ft.
Hence, the answer to the question is that the ball will not rise above 2 ft after 5 bounces.